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Chapter 7: Problem 2

Show that the wave equation \((7.5 .5)\) can be put in the alternative form \((\lambda+2 \mu) \nabla \nabla \cdot e-\mu \nabla \times(\nabla \times \rho)=\rho_{v} \frac{\partial^{2} e}{\partial t^{2}}\) which exhibits directly the parts of the wave equation that vanish for solenoidal and for irrotational waves.

Short Answer

Expert verified
Answer: For solenoidal waves, the displacement vector has zero divergence, thus the first term vanishes: \((\lambda+2 \mu) \nabla \nabla \cdot e = 0\). For irrotational waves, the curl of the displacement vector is zero, hence the second term vanishes: \(\mu \nabla \times(\nabla \times \rho) = 0\).

Step by step solution

01

Identify given equation and desired form

The given wave equation \((7.5 .5)\) should be stated along with desired alternative form: \((\lambda+2 \mu) \nabla \nabla \cdot e - \mu \nabla \times(\nabla \times \rho) = \rho_{v} \frac{\partial^{2} e}{\partial t^{2}}\) **Note**: Here, we assume you missed transcription of actual equation \((7.5 .5)\). However, the provided alternative form and the focus on solenoidal and irrotational waves suggests that we are working with an equation related to the stress tensor in terms of the Lame coefficients (\(\lambda\) and \(\mu\)) and the displacement vector \(e\). If you could provide the original equation stated as \((7.5.5)\), we could better adjust the solution.
02

Analyze properties of solenoidal and irrotational waves

Solenoidal and irrotational waves have distinct properties related to the displacement vector. Solenoidal waves: - For solenoidal waves, the displacement vector has zero divergence, i.e., \(\nabla \cdot e = 0\). Irrotational waves: - For irrotational waves, there is no rotational part in the displacement vector, i.e., the curl is zero, \(\nabla \times e = 0\).
03

Try to rewrite the given equation into desired form

Since we don't know the given equation \((7.5 .5)\), we cannot show the rewriting process accurately. However, if you can provide the exact equation in this form, we can detail step-by-step manipulations using the vector calculus.
04

Show parts of alternative form vanishing for solenoidal and irrotational waves

Assuming the alternative form \((\lambda+2 \mu) \nabla \nabla \cdot e - \mu \nabla \times(\nabla \times \rho) = \rho_{v} \frac{\partial^{2} e}{\partial t^{2}}\) was derived correctly, we can analyze the parts that vanish for solenoidal and irrotational waves. Solenoidal waves: - For solenoidal waves, \(\nabla \cdot e = 0\). This means the first term vanishes: \((\lambda+2 \mu) \nabla \nabla \cdot e = 0\). Irrotational waves: - For irrotational waves, \(\nabla \times e = 0\). This means the second term vanishes: \(\mu \nabla \times(\nabla \times \rho) = 0\). In conclusion, the alternative form directly exhibits the parts of the wave equation that vanish for solenoidal and for irrotational waves.

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Most popular questions from this chapter

Show (a) that there are a maximum of 21 independent elastic constants, using the fact that both stress and strain are symmetric dyadics, and \((b)\) that there exists a symmetry among the constants arising from the reciprocity relations $$ \frac{\partial f_{i j}}{\partial e_{k !}}=\frac{\partial f_{k}}{\partial \epsilon_{i j}} $$ where \(i j\) and \(k l\) stand for pairs of values of \(x, y, z\).

Write the pressure at any depth below the surface of a liquid as a dyadic. Find the force \(d \mathbf{F}\) on an element of area \(d \mathbf{S}\) of a submerged body. Integrate this force over the surface of the body to establish Archimedes' principle. Hint: Use Gauss' theorem for a dyadic $$ \oint d \mathrm{~S} \cdot \mathrm{T}=\int \mathrm{\nabla} \cdot \mathrm{T} d \mathrm{~s} $$ as an aid in evaluating the integrals.

Prove that the three components \(A_{x}=A_{y t}, A_{y}=A_{z_{1}}, A_{z} \equiv A_{x_{y}}\) of an antisymmetric dyadic transform as the components of a vector and hence enable A to be treated as a physical vector field when it is a function of \(x, y, z\), and \(t\).

Show that $$ \lambda+2 \mu=\frac{Y(1-\sigma)}{(1+\sigma)(1-2 \sigma)}=B+\frac{4}{3} \mu . $$ Show that when an external body force per unit volume \(\mathbf{F}_{1 e}\) is present, the equation for the elastic displacement (7.5.5) becomes $$ (\lambda+\mu) \nabla \nabla \cdot e+\mu \nabla \cdot \nabla_{e}+\mathbf{F}_{1 e}=\rho_{\varphi} \frac{\partial^{2} e}{\partial l^{2}} $$ (This equation becomes an equation for elastic equilibrium when \(\varphi\) is not a function of time. Its solution \(\rho(x, y, z)\) must then satisfy specified conditions at the boundary of a body. The stress and strain at each point of the body can be calculated knowing \(\theta\). We thus have formulated the problem of elastic equilibrium in a fundamental way.)

An elastic medium suffers a simple shear in the \(y\) direction such that a point originally at \((x, y)\) moves to the position \((x, y+\gamma x)\), where \(\gamma=\partial_{\eta} / \partial x \ll 1\) is the shearing strain. Examine the transformation of the circle \(x^{2}+y^{2}=a^{2}\) into an ellipse to find the magnitudes and orientation of the principal axes due to a simple shear. \(A\) nswer: To a first order in \(x\) the angles between the principal axes and the \(x\) axis are \((\pi+\gamma) / 4,(3 \pi+\gamma) / 4 ;\) their magnitudes are \(a\left[1 \pm\left(\gamma / 2+\gamma^{2} / 8+\cdots \cdot\right)\right]\).
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